2005-Oujda International Conference on Nonlinear Analysis.
Electronic Journal of Differential Equations, Conference 14, 2006, pp. 119–124.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
FAST AND HETEROCLINIC SOLUTIONS FOR A SECOND
ORDER ODE
MARGARITA ARIAS
Abstract. We present some results on the existence of fast and heteroclinic
solutions of an ODE connected with travelling wave solutions of a FisherKolmogorov’s equation. In particular, we present a variational characterization
of the minimum speed of propagation.
1. Introduction
Some chemical and biological systems can be modelled by an autocatalytic process (see, e.g. [9, 10]). In many of these process the system can support propagating
wavefronts due to a combination of a reaction effect and a molecular diffusion. The
pioneering model in this framework is due to Fisher, [5], who suggested the equation
ut = uxx + u(1 − u)
for studying the spatial spread of a favoured gene in a population. The simplest
generalization of that equation is the so called Fisher-Kolmogorov’s equation
ut = uxx + f (u),
(1.1)
where f is a given function with two zeroes, say u = 0 and u = 1, and positive on
]0, 1[ so that u = 0 and u = 1 are the only two stationary states of (1.1). Equations
like (1.1) arises in many problems suggested, for instance, by the classical theory of
population genetics or by certain flame propagation problems in chemical reactor
theory (see, e.g. [3]).
A travelling wavefront or travelling wave solution (t.w.s., in short) of (1.1) is a
solution u(t, x) having a constant profile, that is, such that
u(t, x) = ϕ(x − ct),
for some fixed ϕ(ξ) (called the wave shape) and a constant c (called the wave speed ).
Specially important for the applications are t.w.s. connecting the two stationary
states, u = 0 and u = 1.
2000 Mathematics Subject Classification. 34C37, 35K57, 49J35.
Key words and phrases. Fisher-Kolmogorov’s equation; travelling wave solutions;
speed of propagation; variational methods; constrained minimum problem.
c
2006
Texas State University - San Marcos.
Published September 20, 2006.
119
120
M. ARIAS
EJDE/CONF/14
A simple calculation shows that if u(t, x) = ϕ(x − ct) is a t.w.s. of (1.1), then
the wave shape ϕ is a solution of the ODE
u00 + cu0 + f (u) = 0.
(1.2)
When a t.w.s. connects the stationary states, its corresponding wave shape is a
positive heteroclinic solution of (1.2) that connects the equilibria 1 and 0, that is,
a solution of (1.2) defined on R and satisfying
u(t) ∈]0, 1[,
∀t ∈ R,
lim u(t) = 1,
t→−∞
lim u(t) = 0.
t→+∞
There is a vast and rich body of literature dealing with the existence of t.w.s.
of (1.1) connecting the stationary states, going from the pioneering work of Kolmogorov, Petrovski and Piskounoff [6], through the remarkable paper of Aronson
and Weinberger [4] up to more recent approaches (see [1, 7, 8, 12]).
It is well known (see, e.g. [4, 7]) that there exists a positive number, c∗ , such
that equation (1.2) has a heteroclinic solution connecting 1 and 0 if and only if
c ≥ c∗ .
In terms of the Fisher-Kolmogorov’s equation, that result says that none t.w.s.
of (1.1) starting from the stationary state u = 1 and moving with speed less than
c∗ reaches the stationary state u = 0. c∗ is called the minimum propagation speed.
It is clear that the heteroclinic solution, if there
exists, is strictly decreasing.
p
When f is differentiable in [0, 1], then c∗ ≥ 2 f 0 (0) since otherwise the origin
cannot acts as an attractor for positive solutions of equation (1.2). It is also proved
(see [4, 7, 1, 12]) that
r
c∗ ≤ 2
sup f (u)/u,
0<u<1
with equality if f is concave in [0, 1].
On the other hand, the way that positive solutions of (1.2) approach zero in
the phase plane, at least when f ∈ C 1 , mimics the phase plane picture for the
corresponding linearization at the origin: There exists an extremal trajectory, Tc ,
in the lower half-plane u0 < 0, that connects some point (1, a), a ≤ 0, with (0, 0)
whose slope at the origin is
p
−c − c2 − 4f 0 (0)
,
λ1 =
2
while every other trajectory in the region u > 0 approaching (0, 0) has slope at the
origin
p
−c + c2 − 4f 0 (0)
λ2 =
.
2
Moreover, Tc is extremal in the sense that trajectories below it stays bounded
away from the origin.
p Aronson and Weinberger (see [4], theorem 4.1) proved that
whenever c∗ > 2 f 0 (0), the extremal trajectory Tc∗ is an heteroclinic solution
between 1 and 0.
This note is a brief summary of the conference given by the author on the ”Colloque International d’Analyse Non linéaire d’Oujda”, about some recent results
obtained in collaboration with J. Campos, A.M. Robles-Pérez and L. Sanchez dealing with some variational problems whose solutions are in correspondence with Tc
and that, in particular, let us give a variational characterization of c∗ . All the
presented results with their proofs can be found in [2].
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FAST AND HETEROCLINIC SOLUTIONS
121
2. A variational characterization of fast solutions
We say that a solution u(t) of equation (1.2) is a fast solution if its corresponding
trajectory is the extremal trajectory Tc . Our purpose is to characterize these solutions in variational terms. In order to do that, we express their speed in approaching
0 by means of an integrability condition:
Given c > 0, we define the space
Z +∞
1
ect u0 (t)2 dt < +∞ and u(+∞) = 0}
Hc := {u ∈ Hloc
(0, +∞) :
0
1/2
. This is a Hilbert space and if u ∈ Hc ,
with the norm kuk = 0 ect u0 (t)2 dt
u obviously tends “quickly” to 0 as t → +∞.
We introduce the functional F : Hc → R defined as
Z +∞
u0 (t)2
F(u) =
ect (
− F (u(t))) dt, u ∈ Hc ,
2
0
Ru
where F (u) := 0 f (s) ds. When
R +∞
(H) f : [0, 1] → R+ is a Lipschitz function such that f (0) = 0 = f (1) and
f (u) > 0 if 0 < u < 1,
one can prove that F is well defined, continuous and in fact differentiable in Hc .
A critical point of F is a solution of equation
(ect u0 )0 + ect f (u) = 0,
(2.1)
or, equivalently, of (1.2). We call it a fast solution because of its integrability
property near +∞.
We prove that a potential minimizer of F in {u ∈ Hc : u(0) = 1} has to verify
0 < u(t) < 1, for all t > 0, and u0 (t) < 0, for all t ≥ 0, and that F has a minimum
2
in {u ∈ Hc : u(0) = 1} provided that there exist 0 < k < c4 with F (u) ≤ ku2 /2,
for all u ∈ [0, 1]. Therefore, we have the following result.
2
Proposition 1. Assume (H) and there exist 0 < k < c4 so that F (u) ≤ ku2 /2, for
all u ∈ [0, 1]. Then equation (1.2) has a fast solution u ∈ Hc defined on t ≥ 0 such
that u(0) = 1 and u0 (t) < 0, for all t ≥ 0.
This result is particularly connected to the existence of heteroclinic solutions.
Indeed, one can prove that
If there exists a solution of (1.2) defined on [0, +∞), with u(0) =
1, u(t) > 0, t > 0 and u(t) → 0 as t → +∞, then equation (1.2)
has an heteroclinic solution.
So, the above proposition proves the existence of heteroclinic solutions whenever
2
≤ c4 , for all u ∈ [0, 1]. Consequently,
2F (u)
u2
c∗ ≤ inf{c > 0 :
2F (u)
c2
≤
,
u2
4
This upper bound generalizes the estimate in [4].
∀u ∈ [0, 1]}.
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M. ARIAS
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3. Fast heteroclinic solutions
After studying the fast solutions, we ask about heteroclinic connections between
the two equilibria u = 1 and u = 0 of equation (1.2). As in the previous section,
we begin by introducing an appropriate space to work.
Given c > 0, we consider the space
Z +∞
1
Xc := {u ∈ Hloc
(R) :
ect u0 (t)2 dt < +∞ and u(+∞) = 0},
−∞
R +∞
1/2
.
with the norm kukc := −∞ ect u0 (t)2 dt
We will say that a solution u, of the equation
u00 + cu0 + λf (u) = 0,
(3.1)
for some λ > 0, is a fast heteroclinic solution if u ∈ Xc and u(−∞) = 1. Note that,
under assumption (H), any heteroclinic connection u(t) of (3.1) between 1 and 0
has the property u(t) ∈]0, 1[, u0 (t) < 0, for all t ∈ R.
Our aim now is to obtain a variational characterization of the smallest value of
λ for which equation (3.1) has a fast heteroclinic solution. We√remark that u(t) is
a solution of (3.1)
√ for some λ > 0 if and only if v(t) := u(t/ λ) is a solution of
(1.2) with c = c/ λ.
To do that, we introduce two real functionals on Xc :
Z +∞
Z +∞
u0 (t)2
Ac (u) :=
dt; Bc (u) :=
ect
ect F (u(t)) dt,
2
−∞
−∞
and we will look for critical points of the restriction of Ac to the set Mc := {u ∈
Xc : Bc (u) = 1}. (Note that Mc is non empty as a consequence of the hypothesis
on f ).
We define
λ(c) := inf{Ac (u) : u ∈ Mc }.
It is easy to check that Ac and Bc are C 1 -functionals and Mc is a C 1 -manifold. By
Lagrange multipliers rule, u ∈ Mc is a critical point of the restriction of Ac to Mc
if and only if u ∈ Mc is a solution of (3.1). Playing appropriately with (3.1) we are
able to prove that
If λ(c) is attained, then equation (3.1) with λ = λ(c) has a fast
heteroclinic solution u ∈ Mc and Ac (u) = λ(c).
Remark 2. Given u ∈ Xc and a ∈ R, the function v(t) := u(t − a) belongs to Xc
and Ac (v) = eca Ac (u), Bc (u) = eca Bc (u). So, if u ∈ Xc is a critical point of Ac
subject to the restriction Bc (u) = 1, for all α > 0, the function v(t) := u(t − lncα )
is a critical point of Ac subject to the restriction Bc (u) = α. Hence, condition
Bc (u) = 1 is a kind of normalization.
The previous result reduces the problem of the existence of fast heteroclinic
solutions to prove that λ(c) is attained. Using a convenient closed convex set, we
show that λ(c) is attained when F (u) = o(u2 ) as u → 0+ .
Finally, working with an auxiliary functional defined on that closed convex set,
we obtain our main result.
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FAST AND HETEROCLINIC SOLUTIONS
123
Theorem 3. Assume (H), and also that there exists f 0 (0) and
λ(c) <
c2
.
4f 0 (0)
(3.2)
Then, λ(c) is attained. In particular, (2.1) with λ = λ(c) has a fast heteroclinic
solution.
Observe that our approach does not require differentiability except at the origin.
On the other hand, if there exists f 0 (0), working with truncations of the function
εe−kt , ε → 0, k ↓ c/2, one can prove
λ(c) ≤
c2
,
4f 0 (0)
and condition (3.2) is almost necessary.
Moreover, as a consequence of this result, if there exists f 0 (0), λ(c) is positive.
A simple change of variable shows that λ(c) = c2 λ(1). Hence, condition (3.2) is
independent of c and it can be write
λ(1) <
1
.
4f 0 (0)
4. A variational characterization of c∗
Theorem 3 let us obtain a variational characterization of the minimum propagation speed c∗ . As we have already said in the introduction,
c∗ := inf{c ∈ R : (1.2) has an heteroclinic solution.}
Mallaguti and Marcelli [8] proved that c∗ is in fact a minimum and it is positive. We
are going to relate this number with the function λ(c) introduced in the previous
section. In order to do that, let us define
1
c̄ := p
.
λ(1)
p
2
Having in mind that λ(c) ≤ 4fc0 (0) and λ(c) = c2 λ(1), one has that c̄ ≥ 2 f 0 (0).
p
From Theorem 3, if c̄ > 2 f 0 (0), equation (1.2) with c = c̄ has a fast heteroclinic
solution and, then, c̄ ≥ c∗ . We can prove the following result.
Theorem 4. c̄ = c∗ .
The proof of this theorem is based on the following result.
p
Proposition 5. Assume that for some c > 2 f 0 (0) there exists an heteroclinic
solution. Then, c = c∗ if and only if this heteroclinic is fast.
Remark that the previous proposition says:
p
At least when c > 2 f 0 (0), c∗ is the only value of the parameter
for which the heteroclinic connection between the two equilibria of
(1.2) is fast.
The proof of this result follows by interpreting positive decreasing solutions of (1.2)
as solutions of a suitable first order equation (as it has been done in [11, 7]).
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M. ARIAS
EJDE/CONF/14
A positive decreasing solution of (1.2) has a trajectory in the second quadrant of
the (u, u0 )-plane. It is about looking at such a trajectory as the graph of a function
φ, so that u0 = φ(u). Putting y(u) = φ(u)2 , y is a solution of
dy
√
(4.1)
= 2c y − 2f (u).
du
A heteroclinic solution of (1.2) corresponds to a positive solution of (4.1) on ]0, 1[
such that y(0) = y(1) = 0.
(Note that the Cauchy problem for equation (4.1) has no uniqueness, but any
solution of (4.1) can be continued as long as it remains positive.)
Summarizing, we obtain
Z +∞
n Z +∞
o−1
u0 (t)2
c∗ = inf
ect
dt : u ∈ X1 ,
ect F (u(t)) dt = 1
.
2
−∞
−∞
Moreover, when c > c∗ equation (1.2) has an heteroclinic connection between its
equilibria though it is no
p fast, that is, the extremal trajectory Tc is not an heteroclinic, but if c =pc∗ > 2 f 0 (0), then Tc connects the two equilibria.
When c∗ = 2 f 0 (0), (1.2) has an heteroclinic connection between its equilibria,
but it is an open problem to know if it is or not a fast heteroclinic.
References
[1] S. Ahmad and A.C. Lazer, An elementary approach to travelling front solutions to a system
of N competition-diffusion equations, Nonlinear Anal. TMA 16, 893-901 (1991).
[2] M. Arias, J. Campos, A.M. Robles-Pérez and L. Sanchez, Fast and heteroclinic solutions for a second order ODE related to Fisher-Kolmogorov’s equation., Calc. Var. Partial
Differential Equations 21, no. 3, 319–334, (2004).
[3] D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, commbustion, and nerve propagation, in ”Partial Differential Equations and Related Topics, Lecture Notes in Mathematics”,Vol. 446, pp. 5-49, Springer, New York, 1975.
[4] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30, 33-76 (1987).
[5] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 353-369 (1937).
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[8] L. Malaguti and C. Marcelli, Travelling wavefronts in reaction-diffusion equations with
convection effects and non-regular terms, Math. Nachr., 242, 148-164 (2002).
[9] P.M. Mcabe, J.A. Leach and D.J. Needham, The evolution of travelling waves in fractional
order autocatalysis with decay. I. Permanent form travelling waves, SIAM J. Appl. Math.,
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[10] J.D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
[11] P.L. Sachdev, Nonlinear ordinary differential equations and their applications, M. Dekker,
N. York 1991.
[12] L. Sanchez, A note on a nonautonomous O.D.E. related to the Fisher equation, J. Comp.
Appl. Math. 113, 201-209 (2000).
Margarita Arias
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain
E-mail address: marias@goliat.ugr.es